How many times can I split a coaxial cable.
I work for a small-ish ISP (~100,000 users) and we support a almost 100 seperate cable systems. Only one of them is cable modem internet, the rest are purely TV service, but the low level of getting enough signal to each connected device (TV, modem) is the same. I have been involved in several new installation projects, both in planning and installing the hardware. With this experience, I would barely be able to plan a new system on my own. While it is good that you are curious and want to make a new service for your community, I highly advice you to at the very least hire a cable engineer as a consultant.
That being said, I will try to answer the original question. From a downstream signal perspective (that is, from the CMTS to the modems) you can basically split it as many times as you want, with good quality splitters and amplifiers. You'll need a lot of amplifiers and/or powered splitters though. From an upstream perspective, the amplifiers should do reverse amplification of the low frequency signals used for upload, but there tends to be more of a limit sonce the cable modems output a weaker signal than the CMTS.
You don't want to have 200 modems on one cable network. When we set up our one location with cable modem service, we were recommended to put only 100 modems per CMTS. That means you would have two seperate cable networks in your plan, and I would actually split it into three. When people talk about congestion in the cable network for the neighborhood, this is what they are talking about.
Unless these 200 modems are all in the same building, you're going to end up needing at least one powered outdoor amplifier, and maybe outdoor splitters. Outdoor cable equipment (and some indoor) is always powered by DC voltage injected into the cable system, This is usually 48V DC but is sometimes 60V or higher. THIS CAN EASILY KILL YOU BECAUSE IT IS DC. AC IS ACTUALLY SAFER. TRAINED AND EXPERIENCED CABLE AND TELEPHONE ENGINEERS ARE REGULARLY HURT IF THEY OR SOMEONE ELSE MAKES A MISTAKE. ALWAYS ENSURE THE CABLE YOU ARE WORKING ON IS NOT ENERGIZED. Because of this, ALL outside cable plants should be have the proper permits and be installed by trained professionals.
(technically it is amperage that kills, not voltage. Higher voltages make it easier for the electrons to jump through your body. DC is more dangerous than AC because it makes your muscles lock up, possibly locking your hand onto the wire or whatever else gave it the shock. AC makes your muscles spasm, so you will usually break contact with the voltage source almost immediately.)
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